Questions and Answers of Introduction To Probability Statistics

If the mean annual compensation paid to the chief executives of three engineering firms is $\$ 175,000$, can one of them receive $\$ 550,000$ ?
Records show that the normal daily precipitation for each month in the Gobi desert, Asia is 1, 1, 2, 4, 7 , 15, 29, 27, 10, 3, 2 and $1 \mathrm{~mm}$. Verify that the mean of these figures is 8.5 and
The output of an instrument is often a waveform. With the goal of developing a numerical measure of closeness, scientists asked 11 experts to look at two waveforms on the same graph and give a number
With reference to the preceding exercise, find $s$ using(a) the formula that defines $s$;(b) the handheld calculator formula for $s$.
Meat products are regularly monitored for freshness. A trained inspector selects a sample of the product and assigns an offensive smell score between 1 and 7 where 1 is very fresh. The resulting
With reference to Exercise 2.31, find $s^{2}$ using(a) the formula that defines $s^{2}$;(b) the handheld calculator formula for $s^{2}$.Data From Exercise 2.31 Data From Exercise 1.8Data From Table
For the five observations 8 2 10 6 9(a) calculate the deviations $\left(x_{i}-\bar{x}\right)$ and check that they add to 0 .(b) calculate the variance and the standard deviation.
With reference to Exercise 2.14 on page 34, draw a boxplot.Data From Exercise 2.14 2.14 An engineer uses a thermocouple to monitor the tem- perature of a stable reaction. The ordered values of 50
A factory experiencing a board-solder defect problem with an LED panel board product tested each board manufactured for LED failure. Data were collected on the area of the board on which LEDs were
Refer to Exercise 2.44. The measurements for the 8 panels that did not fail were Good $33.5 \quad32.25\quad34.75\quad34.25\quad35.5\quad33.0\quad36.25\quad 35.75$(a) Calculate the sample mean
Use the distribution in Exercise 2.10 on page 32 to find the mean and the variance of the nanopillar diameters.Data From Exercise 2.10 2.10 To continually increase the speed of computers, elec-
Use the distribution obtained in Exercise 2.12 on page 33 to find the mean and the standard deviation of the particle sizes. Also determine the coefficient of variation.Data From Exercise 2.12 2.12
Use the distribution obtained in Exercise 2.14 on page 33 to find the coefficient of variation of the temperature data.Data From Exercise 2.14 2.14 An engineer uses a thermocouple to monitor the tem-
Show that\[\sum_{i=1}^{n}\left(x_{i}-\bar{x}\right)=0\]for any set of observations $x_{1}, x_{2}, \ldots, x_{n}$.
Show that the computing formula for $s^{2}$ on page 44 is equivalent to the one used to define $s^{2}$ on page 37.
If data are coded so that xic uia, show that cu+a and sx |C| Su =
Median of grouped data To find the median of a distribution obtained for $n$ observations, we first determine the class into which the median must fall. Then, if there are $j$ values in this class
For each of the following distributions, decide whether it is possible to find the mean and whether it is possible to find the median. Explain your answers.(a)(b)IQFrequencyless than
To find the first and third quartiles $Q_{1}$ and $Q_{3}$ for grouped data, we proceed as in Exercise 2.53, but count $\frac{n}{4}$ and $\frac{3 n}{4}$ of the observations instead of
If $k$ sets of data consist, respectively, of $n_{1}, n_{2}, \ldots, n_{k}$ observations and have the means $\bar{x}_{1}, \bar{x}_{2}, \ldots, \bar{x}_{k}$, then the overall mean of all the data is
The formula for the preceding exercise is a special case of the following formula for the weighted mean:\[\bar{x}_{w}=\frac{\sum_{i=1}^{k} w_{i} x_{i}}{\sum_{i=1}^{k} w_{i}}\]where $w_{i}$ is a
Modern computer software programs have come a long way toward removing the tedium of calculating statistics. MINITAB is one common and easy-to-use program. We illustrate the use of the computer using
With the observations on the strength (in pounds per square inch) of $2 \times 4$ pieces of lumber already set in $\mathrm{C} 1$, the sequence of choices and clicks produces an even more complete
From 1,500 wall clocks inspected by a manufacturer, the following defects were recorded.hands touch each other112defective gears 16faulty machinery 18rotating pin 6others3Create a Pareto chart.
Create(a) a frequency table of the aluminum alloy strength data on page 29 using the classes [66.0, 67.5), $[67.5,69.0),[69.0,70.5),[70.5,72.0),[72.0$, $73.5),[73.5,75.0),[75.0,76.5)$, where the
Create(a) a frequency table of the inter-request time data on page 29 using the intervals $[0,2,500)$, $[2,500,5,000),[5,000,10,000),[10,000,20,000)$, $[20,000,40,000), \quad[40,000,60,000),
Direct evidence of Newton's universal law of gravitation was provided from a renowned experiment by Henry Cavendish (1731-1810). In the experiment, masses of objects were determined by weighing, and
J. J. Thomson (1856-1940) discovered the electron by isolating negatively charged particles for which he could measure the mass/charge ratio. This ratio appeared to be constant over a wide range of
With reference to Exercise 2.64,(a) calculate the median, maximum, minimum, and range for Tube 1 observations;(b) calculate the median, maximum, minimum, and range for the Tube 2 observations.Data
A. A. Michelson (1852-1931) made many series of measurements of the speed of light. Using a revolving mirror technique, he obtained$\begin{array}{lllllllllll}12 & 30 & 30 & 27 & 30 & 39 & 18 & 27 &
With reference to Exercise 2.66,(a) find the quartiles;(b) find the minimum, maximum, range, and interquartile range;(c) create a boxplot.Data From Exercise 2.66 2.66 A. A. Michelson (1852-1931)
An electric engineer monitored the flow of current in a circuit by measuring the flow of electrons and the resistance of the medium. Over 11 hours, she observed a flow of\[\begin{array}{lllllllllll}5
With reference to Exercise 2.68,(a) find the quartiles;(b) find the minimum, maximum, range, and interquartile range;(c) construct a boxplot.Data From Exercise 2.68 2.68 An electric engineer
The weight (grams) of meat in a pizza product produced by a large meat processor is measured for a sample of $n=20$ packages. The ordered values are (Courtesy of Dave
With reference to Exercise 2.70, construct(a) a boxplot.(b) a modified boxplot.Data From Exercise 2.70 2.70 The weight (grams) of meat in a pizza product pro- duced by a large meat processor is
During the laying of gas pipelines, the depth of the pipeline (in $\mathrm{mm}$ ) must be controlled. One service provider recorded depths of418428431420412425423433417420410431429425(a) Find the
With reference to the lumber-strength data in Exercise 2.59, the statistical software package $S A S$ produced the output in Figure 2.20. Using this output,(a) identify the mean and standard
An engineer was assigned the task of calculating the average time spent by vehicles waiting at trafficsignals. The signal timing (in seconds) would then be modified to reduce the pressure of traffic.
The National Highway Traffic Safety Administration reported the relative speed (rounded to the nearest $5 \mathrm{mph}$ ) of automobiles involved in accidents one year. The percentages at different
Given a five-number summary,is it possible to determine whether or not an outlier is present? Explain. minimum Q1 Q2 Q3 maximum
Given a stem-and-leaf display, is it possible to determine whether or not an outlier is present? Explain.
Traversing the same section of interstate highway on 11 different days, a driver recorded the number of cars pulled over by the highway patrol:\[\begin{array}{lllllllllll}0 & 1 & 3 & 0 & 2 & 0 & 1 &
An experimental study of the atomization characteristics of biodiesel fuel ${ }^{5}$ was aimed at reducing the pollution produced by diesel engines. Biodiesel fuel is recyclable and has low emission
Show the set [0, 1) is uncountable. That is you can never provide a list in the form of {a1, a2, a3,⋯} that contains all the elements in [0, 1).
A certain disease affects about 1 out of 10, 000 people. There is a test to check whether the person has the disease. The test is quite accurate. In particular, we know thatThe probability that the
I roll a die n times, n ∈ N. Find the probability that numbers 1 and 6 are both observed at least once.
Consider a c ommunication system. At any given time, the communication channel is in good condition with probability 0.8, and is in bad condition with probability 0.2. An error occurs in a
A box contains two coins: a regular coin and one fake two-headed coin (P(H) = 1). I choose a coin at random and toss it twice. Define the following events.A= First coin toss results in an H.B= Second
Let X be a discrete random variable with the following PMF:a. Find RX, the range of the random variable X.b. Find P(X ≥ 1.5).c. Find P(0 d. Find P(X = 0|X Px(x)= T | 16 2 1 3 0 for x = 0 for x =
In a factory th ere are 100 units of a certain product, 5 of which are defective. We pick three units from the 100 units at random. What is the probability that exactly one of them is defective?
50 students live in a dormitory. The parking lot has the capacity for 30 cars. If each student has a car with probability 1/2 (independently from other students), what is the probability that there
Let X be a discrete random variable with the following PMF:Find and plot the CDF of X. Px(x) 0.2 0.3 0.2 0.2 0.1 0 for x = = -2 -1 for x = for x = 0 for x = 1 for x = 2 otherwise
I toss a coin twice. Let X be the number of observed heads. Find the CDF of X.
Let X ∼ Geometric(p). Find V ar(X).
Let X be a mixed random variable with the following generalized PDFa. Find P(X = 1) and P(X = −2).b. Find P(X ≥ 1).c. Find P(X = 1|X ≥ 1).d. Find EX and V ar(X). fx(x) = (2+2) + - (1 - (-1)
Consider two random variables X and Y with joint PMF given in Table 5.4 Joint PMF of X and Y a. Find P(X ≤ 2,Y > 1).b. Find the marginal PMFs of X and Y .c. Find P(Y = 2|X = 1).d. Are X and Y
N people sit around a round table, where N > 5. Each person tosses a coin. Anyone whose outcome is different from his/her two neighbors will receive a present. Let X be the number of people who
Prove the union bound using Markov's inequality.
50 students live in a dormitory. The parking lot has the capacity for 30 cars. Each student has a car with probability 1/2, independently from other students. Use the CLT (with continuity correction)
I have a bag that contains 3 balls. Each ball is either red or blue, but I have no information in addition to this. Thus, the number of blue balls, call it θ, might be 0, 1, 2, or 3. I am allowed to
Suppose that the random variable X is transmitted over a communication channel. Assume that the received signal is given by Y = 2X +W, where W ∼ N(0,σ2) is independent of X. Suppose that X = 1
Suppose that the number of customers visiting a fast food restaurant in a given time interval I is N ∼ Poisson(μ). Assume that each customer purchases a drink with probability p, independently
Let N(t) be a Poisson process with rate λ. Let 0 < s < t. Show that given N(t) = n, N(s) is a binomial random variable with parameters n and p = s/t.
Let Xn be a discrete-time Markov chain. Remember that, by definition, p(n)ii = P(Xn = i|X0 = i). Show that state i is recurrent if and only if (η) Σ Pu = ∞. n=1
Write R programs to generate Geometric(p) and Negative Binomial(i,p) random variables.
Use the algorithm for generating discrete random variables to obtain a Poisson random variable with parameter λ = 2.
Explain how to generate a random variable with the densityif your random number generator produces a Standard Uniform random variable U. f(x) = 2.5x√x for 0
Use the inverse transformation method to generate a random variable having distribution function F(x) x² + x 2 0 ≤ x ≤ 1
Let X have a standard Cauchy distribution.Assuming you have U ∼ Uniform(0; 1), explain how to generate X. Then, use this result to produce 1000 samples of X and compute the sample mean. Repeat the
When we use the Inverse Transformation Method, we need a simple form of the cdf F(x) that allows direct computation of X = F-1(U). When F(x) doesn't have a simple form but the pdf f(x) is available,
Use the rejection method to generate a random variable having density function Beta(2; 4). Assume g(x) = 1 for 0 < x < 1.
Use the rejection method to generate a random variable having the Gamma( 5/2 ; 1) density function.Assume g(x) is the pdf of the Gamma (a = /2, λ = 1).
Use the rejection method to generate a standard normal random variable. Assume g(x) is the pdf of the exponential distribution with λ = 1.
Use the rejection method to generate a Gamma(2; 1) random variable conditional on its value being greater than 5, that isAssume g(x) be the density function of exponential distribution. f(x)
Let X and Y be jointly normal and X ∼ N(0, 1), Y ∼ N(1, 4), and ρ(X,Y ) = 1/2. Find a 95% credible interval for X, given Y = 2 is observed.
Assume our data Y given X is distributed Y | X = x ∼ Binomial(n, p = x) and we chose the prior to be X ∼ Beta(α,β). Then the PMF for our data isand the PDF of the prior is given byNote that,a.
Find the average error probability in Problem 13.Problem 13Suppose that the random variable X is transmitted over a communication channel. Assume that the received signal is given by Y = 2X +W, where
When the choice of a prior distribution is subjective, it is often advantageous to choose a prior distribution that will result in a posterior distribution of the same distributional family. When the
A monitoring system is in charge of detecting malfunctioning machinery in a facility. There are two hypotheses to choose from:H0: There is not a malfunction,H1: There is a malfunction.The system
Let X and Y be jointly normal and X ∼ N(2, 1), Y ∼ N(1, 5), and ρ(X,Y ) = 1/4. Find a 90% credible interval for X, given Y = 1 is observed.
Assume our data Y = (y1, y2,…, yn)T given X is independently identically distributed, Y | X = x i.i.d. ∼ Exponential(λ = x), and we chose the prior to be X ∼ Gamma(α,β).a. Find the
Assume our data Y given X is distributed Y | X = x ∼ Geometric(p = x) and we chose the prior to be X ∼ Beta(α,β). Refer to Problem 18 for the PDF and moments of the Beta distribution.a. Show
Let {Xn,n ∈ Z} be a discrete-time random process, defined aswhere Φ ∼ Uniform(0, 2π).a. Find the mean function, μX(n).b. Find the correlation function RX(m,n).c. Is Xn a WSS process? (+), Xn =
You have 1000 dollars to put in an account with interest rate R, compounded annually. That is, if Xn is the value of the account at year n, thenThe value of R is a random variable that is determined
Let {X(t), t ∈ R} be a continuous-time random process, defined aswhere A ∼ U(0, 1) and Φ ∼ U(0, 2π) are two independent random variables.a. Find the mean function μX(t).b. Find the
Let {X(n),n ∈ Z} be a WSS discrete-time random process with μX(n) = 1 and RX(m,n) = e−(m−n)2. Define the random process Z(n) asa. Find the mean function of Z(n), μZ(n).b. Find the
Let {X(t), t ∈ [0,∞)} be defined as X(t) = A +Bt, for all t ∈ [0,∞),where A and B are independent normal N(1, 1) random variables.a. Find all possible sample functions for this random
Let g : R ↦ R be a periodic function with period T, i.e.,Define the random process {X(t), t ∈ R} aswhere U ∼ Uniform(0,T). Show that X(t) is a WSS random process. g(t+T)= g(t), for all t = R.
Consider the random process {Xn,n = 0, 1, 2,⋯}, in which Xi's are i.i.d. standard normal random variables.1. Write down fXn(x) for n = 0, 1, 2,⋯.2. Write down fXmXn(x1,x2) for m ≠ n.
Find the mean functions for the random processes given in Examples 10.1 and 10.2.Examples 10.1You have 1000 dollars to put in an account with interest rate R, compounded annually. That is, if Xn is
Let {X(t), t ∈ R} and {Y (t), t ∈ R} be two independent random processes. Let Z(t) be defined asProve the following statements:a. μZ(t) = μX(t)μY (t), for all t ∈ R.b. RZ(t1, t2) = RX(t1,
Find the correlation functions and covariance functions for the random processes given in Examples 10.1 and 10.2.Examples 10.1You have 1000 dollars to put in an account with interest rate R,
Let A, B, and C be independent normal N(1, 1) random variables. Let {X(t), t ∈ [0,∞)} be defined asAlso, let {Y (t), t ∈ [0,∞)} be defined asFind RXY (t1, t2) and CXY (t1, t2), for t1, t2 ∈
Let X(t) be a Gaussian process such that for all t > s ≥ 0 we have X(t) −X(s) ∼ N (0, t −s).Show that X(t) is mean-square continuous at any time t ≥ 0.
Let X(t) be a WSS Gaussian random process with μX(t) = 1 and RX(τ) = 1 +4sinc(τ)a. Find P(1 < X(1) < 2).b. Find P(1 < X(1) < 2,X(2) < 3).
Let X(t) be a zero-mean WSS Gaussian process with RX(τ) = e−τ 2 , for all τ ∈ R.1. Find P(X(1) < 1).2. Find P(X(1) +X(2) < 1).
Consider a WSS random process X(t) withwhere a is a positive real number. Find the PSD of X(t). Rx(T) = e-a|¹|,
Let X(t) be a Gaussian random process with μX(t) = 0 and RX(t1, t2) = min(t1, t2).Find P(X(4) < 3|X(1) = 1).
Let {X(t), t ∈ R} be a continuous-time random process, defined aswhere A0, A1, ⋯, An are i.i.d. N(0, 1) random variables and n is a fixed positive integer.a. Find the mean function μX(t).b. Find
Let X(t) be a zero-mean WSS process with RX(τ) = e−|τ|. X(t) is input to an LTI system withLet Y (t) be the output.a. Find μY (t) = E[Y(t)].b. Find RY (τ).c. Find E[Y (t)2]. |H(f)|

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